Preheat the oven to 350 degrees F (175 degrees C).
Combine oats and water in a medium bowl. Let sit for 20 minutes.
Combine brown sugar, 1/2 cup butter, white sugar, eggs, and vanilla extract in a large bowl. Add oatmeal mixture and mix until combined.
Sift flour, baking soda, 1 teaspoon cinnamon, nutmeg, salt, and allspice together in a bowl. Add to oatmeal mixture.
Place 2 tablespoons butter in a 9-inch square baking pan and melt in the preheated oven. Remove from the oven and sprinkle sugar and cinnamon on top. Sprinkle plums evenly over sugar. Pour batter over plums and spread evenly.
Bake in the preheated oven until a toothpick inserted in the center comes out clean, about 50 minutes.